turkmath.org

I shall briefly describe my reflections on the Langlands reciprocity and functoriality principles.

Let $K$ be a number field. The local Langlands group $L_{K_\nu}$ of $K_\nu$ is defined by $L_{K_\nu}=WA_{K_\nu}=W_{K_\nu}\times\mathsf{SU}(2)$ if $\nu\in\mathbb h_K$, and by $L_{K_\nu}=W_{K_\nu}$ if $\nu\in\mathbb a_K$, where $W_{K_\nu}$ denotes the local Weil group of $K_\nu$. For each $\nu\in\mathbb h_K$, fix a Lubin-Tate splitting $\varphi_{K_\nu}$. The local non-abelian norm residue symbol
\begin{equation*}
\{\bullet,K_\nu\}_{\varphi_\nu}^{\mathrm{Langlands}}:{}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}\times
\mathsf{SU}(2)\xrightarrow{\sim}L_{K_\nu}
\end{equation*}
of $K_\nu$ "in the sense of Langlands'' has been defined and studied in the papers of the speaker, where
${}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}$ is a certain non-commutative topological group depending only on the ground field $K_\nu$ and constructed, using Fontaine-Wintenberger theory of fields of norms.

Fix $\underline{\varphi}=\{\varphi_{K_\nu}\}_{\nu\in\mathbb h_K}$ and define a non-commutative topological group $\mathscr {WA}_K^{\underline{\varphi}}$ depending only on the ground field $K$ by the "restricted free topological product''
\[
\mathscr {WA}_K^{\underline{\varphi}}:=
{\ast_{\nu\in\mathbb h_K}}'
\left({}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}\times\mathsf{SU}(2):
{}_1{\nabla_{K_\nu}^{(\varphi_{K_\nu})}}^{\underline 0}\times\mathsf{SU}(2)
\right)\ast
W_\mathbb R^{\ast r_1}\ast W_\mathbb C^{\ast r_2}.
\]
Here, $r_1$ and $r_2$ denote the numbers of real and pairs of complex conjugate embeddings of the number field $K$ in $\mathbb C$. Note that, ${\mathscr {WA}_K^{\underline{\varphi}}}^{ab}=\mathbb J_K$, the idèle group of $K$. So, $\mathscr {WA}_K^{\underline{\varphi}}$ can be viewed as a non-abelian generalisation of $\mathbb J_K$.

Let $L_K$ denote the hypothetical Langlands group $L_K$ of $K$. The existence problem of $L_K$ is one of the major conjectures in the Langlands Program, and according to Arthur, this conjecture is the most fundamental and mysterious one.

For $\nu\in\mathbb h_K\cup\mathbb a_K$, an embedding  $e_\nu:K^{sep}\hookrightarrow K_\nu^{sep}$ determines a continuous homomorphism $e_\nu^{\mathrm{Langlands}}:L_{K_\nu}\rightarrow L_K$ unique up to conjugacy, which in return defines a continuous homomorphism
\[
\mathsf{NR}_{K_\nu}^{(\varphi_{K_\nu})^{\mathrm{Langlands}}}:
{}_\mathbb Z\nabla_{K_\nu}^{\varphi_{K_\nu}}\times\mathsf{SU}(2)
\xrightarrow[\sim]{\{\bullet,K_\nu\}_{\varphi_{K_\nu}}\times
\mathrm{id}_{\mathsf{SU}(2)}}
L_{K_\nu}\xrightarrow{e_\nu^{\mathrm{Langlands}}} L_K
\]
unique up to conjugacy, for each $\nu\in\mathbb h_K$. Fixing one such morphism for each $\nu\in\mathbb h_K$, the collection $\left\{\mathsf{NR}_{K_\nu}^{(\varphi_{K_\nu})^{\mathrm{Langlands}}}\right\}_{\nu\in\mathbb h_K}$ defines a continuous homomorphism
\[\mathsf{NR}_K^{\underline\varphi^{\mathrm{Langlands}}}:\mathscr {WA}_K^{\underline{\varphi}}\rightarrow L_K,\]
unique up to "local conjugation'', called the global non-abelian norm residue symbol of $K$ "in the sense of Langlands'', which is also compatible with Arthur's proposed construction of $L_K$. The key philosophical observation is that the source $\mathscr {WA}_K^{\underline{\varphi}}$ is an unconditional object while the target $L_K$ is conjectural!

Let $\mathrm{G}$ be a connected, quasisplit reductive group over $K$. There is a bijection between the set of "$WA$-parameters''
\[\phi:\mathscr {WA}_K^{\underline{\varphi}}\rightarrow {}^L\mathrm{G}(\mathbb C)=
\widehat{\mathrm G}(\mathbb C)\rtimes L_K\]
of $\mathrm{G}$ over $K$ and the set $\mathscr P_{\mathrm G}$ whose elements are the collections
\[\{\phi_\nu:L_{K_\nu}\rightarrow{}^L\mathrm{G}_\nu(\mathbb C)\}_{\nu\in\mathbb h_K\cup\mathbb a_K}\]
consisting of local $L$-parameters of $\mathrm{G}_\nu$ over $K_\nu$ for each $\nu$. Note that, assuming the local reciprocity principle for $\mathrm{G}_\nu$ over $K_\nu$ for all $\nu\in\mathbb h_K\cup\mathbb a_K$, the set  $\mathscr P_{\mathrm G}$ is in bijection with the set whose elements are the collections
$\{\Pi_{\phi_\nu}\}_{\nu\in\mathbb h_K\cup\mathbb a_K}$  of local $L$-packets of $\mathrm{G}_\nu$ over $K_\nu$ for each $\nu$. As global admissible $L$-packets of $\mathrm G$ over $K$ are the restricted tensor products of local $L$-packets of $\mathrm G_\nu$ over $K_\nu$, by Flath's decomposition theorem, we end up having the following theorems
Theorem.
Let $\mathrm{G}$ be a connected quasisplit reductive group over the number field $K$. Assume that the local Langlands reciprocity principle for $\mathrm{G}_\nu$ over $K_\nu$ for all $\nu\in\mathbb h_K\cup\mathbb a_K$ holds. Then, there exists a bijection
\[
\{\text{$WA$-parameters of $\mathrm G$ over $K$}\}\leftrightarrow\{\text{global admissible $L$-packets of
$\mathrm G$ over $K$}\}
\]
satisfying the "naturality'' properties.

and

Theorem.
Let $\mathrm{G}$ and $H$ be connected quasisplit reductive groups over the number field $K$. Let
\[\rho:{}^LG\rightarrow {}^LH\]
be an $L$-homomorphism.Assume that the local Langlands reciprocity principle for $\mathrm G$ over $K$ holds. Then, the $L$-homomorphism $\rho:{}^LG\rightarrow {}^LH$ induces a map (lifting) from the global admissible $L$-packets of $G$ over $K$ to the global admissible $L$-packets of $H$ over $K$  satisfying the "naturality'' properties.